ACTIVE 200618-20 SEPTEMBER 2006ADELAIDE AUSTRALIA

A high-authority/low-authority control strategy for coupledaircraft-style bays

N. H. Schillera, C. R. FullerVirginia Tech

Blacksburg, VAUnited States, 24061

R. H. CabellNASA Langley Research Center

Hampton, VAUnited States, 23681

ABSTRACTThis paper presents a numerical investigation of an active structural acoustic con-trol strategy for coupled aircraft-style bays. While structural coupling can destabilizeor limit the performance of some model-based decentralized control systems, fully-coupled centralized control strategies are impractical for typical aircraft containingseveral hundred bays. An alternative is to use classical rate feedback with matched,collocated transducer pairs to achieve active damping. Unfortunately, due to the con-servative nature of this strategy, stability is guaranteed at the expense of achievablenoise reduction. Therefore, this paper describes the development of a combined con-trol strategy using robust active damping in addition to a high-authority controllerbased on linear quadratic Gaussian (LQG) theory. The combined control system isevaluated on a tensioned, two-bay model using piezoceramic actuators and ideal pointvelocity sensors. Transducer placement on the two-bay structure is discussed, and theadvantages of a combined control strategy are presented.

1 INTRODUCTION

Active control strategies designed to reduce turbulent boundary layer (TBL) induced noisein commercial and general aviation aircraft have received a lot of attention in the pastdecade. Since boundary layer noise is essentially random in space and time, a coherentreference signal is often unavailable. Therefore, a common approach is to feed back sig-nals from accelerometers or piezoelectric transducers to piezoceramic actuators integratedin the structure [1]. One particularly simple and robust control strategy uses a distributedarray of actuator/sensor pairs with local feedback loops, as described by Elliott et al. [2]. Ifthe transducer pairs are collocated and dual, then any passive (energy dissipative) controllaw, such as negative rate feedback, will guarantee unconditional stability of the closed-loop system [3]. Unfortunately, real transducer pairs are never perfectly collocated anddual, which eliminates the passive property of the system at high frequencies. Despite thislimitation, researchers have shown promising results using real transducer pairs [4, 5, 6].Since this approach essentially augments the inherent damping in the structure, it is wellsuited for lightly damped metallic structures such as the aluminum sidewall of an aircraft.However, the conservative nature of this technique provides the stability guarantees atthe expense of performance. Therefore, it has been referred to as a low-authority control(LAC) approach [7].

In contrast, high-authority control (HAC) approaches offer potential performance ben-efits, as shown by Clark and Cox [8], at the expense of controller robustness. Since HACapproaches, such as linear quadratic Gaussian (LQG) control, are model based, the per-formance of the controller depends on the fidelity of the model [9]. Poorly modeled dy-namics can cause spillover or destabilize the closed-loop system. Spillover is the unde-

aEmail address:nschille@vt.edu

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https://ntrs.nasa.gov/search.jsp?R=20060048499 2018-06-15T17:54:00+00:00Z

sired amplification of the response with respect to the open-loop system. Despite thislimitation, researchers have experimentally shown that HAC approaches can be used ef-fectively to reduce the radiated sound power from simple structures [10, 11, 12]. Unfortu-nately, transitioning decentralized HAC systems from single bay structures to more real-istic multiple-bay structures is difficult, as explained by Gibbs and Cabell [13]. Althougha fully-coupled, centralized control strategy avoids some of the problems associated withdecentralized control, it is not practical for large multiple-bay systems. Since the goalof this work is to develop a control system that can be scaled up with little additionalcomplexity, centralized control strategies are not discussed in this paper.

A high-authority/low-authority control (HAC/LAC) architecture, originally developedto control large flexible space structures, can be used to combine the robustness benefits ofcollocated rate feedback with the performance benefits of modern control strategies [7, 14,15, 16]. The HAC portion of the controller is typically designed to meet the performanceobjective, while the LAC loops add damping and reduce the spillover problems associatedwith HAC of lightly damped systems.

A combined HAC/LAC strategy designed to reduce the sound radiation from aircraft-style panels could be particularly beneficial if a partially correlated and causal referencesignal was available. While TBL noise is essentially random in space and time, a com-ponent of jet noise called shockcell noise is spatially correlated with a broad spectralpeak [17]. If a reference signal correlated with shockcell noise was available, then a feed-forward high-authority approach could be combined with low-authority local feedbackloops. This type of combined control system could provide overall performance benefitsfor mixed disturbances, such as TBL and shockcell noise.

However, for this paper we assume that no reference signal is available, and presenta combined control strategy using LQG control and active damping. The paper beginswith a description of the coupled two-bay model used for these simulations. Next, resultsare presented for a decentralized LQG system, along with some of the advantages anddisadvantages of that approach. Three LAC configurations are then discussed, followedby the combined HAC/LAC system.

2 SYSTEM STUDIED

This analysis considers a thin aluminum panel, 508 mm×536 mm×1.6 mm, partitionedinto two bays by a stiffener, as shown in Figure1. The panel represents a section of anaircraft’s fuselage, which is set between rigid ring frames and stiffened by a stringer. Forthis analysis, the stiffener is offset from the middle of the panel by 11 mm to make thetwo bays different sizes. In addition, all edges of the stiffened panel are assumed to beclamped, and in-plane tension is included to simulate the hoop stress in a pressurizedfuselage.

Analytical models of relatively simple stiffened panels can be created using transfermatrix or finite element-strip methods [18, 19]. These techniques have been used to an-alyze two-dimensional structures when the eigenfunctions in one direction are alreadyknown. A more general approach is to perform a normal modes analysis of the structureusing numerical finite element methods. For this work, a finite element model of the stiff-ened panel was created using MSC.NASTRAN. A normal modes analysis was then used

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Figure 1: Diagram of the tensioned, stiffened panel.

to generate the structural mass and stiffness matrices and to determine the eigenpropertiesof the first 60 modes of the panel. The first nine modes of the stiffened panel are shownin Figure2. Notice that the model includes both global and local modes. The local modesare a result of the size difference between the two bays.

The investigation presented here is limited to piezoceramic actuators and ideal pointvelocity sensors. While the mass of the sensors is neglected, the dynamics of the piezo-electric actuators are included in the system model as,

[Ms + Mp] η + Csη + [Ks + Kp] η = Bff + Θv (1)

whereη is a vector of generalized displacements,Ms is the structural mass matrix,Mp

is the piezoceramic mass matrix,Cs is the structural damping matrix,Ks is the structuralstiffness matrix,Kp is the piezoceramic stiffness matrix,Bf is the forcing matrix,Θ isthe electromechanical coupling matrix,f is a vector of structural forces, andv is a vectorof applied voltages [1, 10]. For this investigation, we assume proportional damping witha modal damping ratio of 0.01 for all 60 modes included in the model.

Although the structural mass and stiffness matrices could be determined directly fromthe FE model, the piezoelectric mass, stiffness, and electromechanical coupling matriceshad to be calculated using,

Mp =

∫χ

ΦT Sp(χ)ρpΦdχ (2)

Kp =

∫χ

(LuΦ)T Sp(χ)κp(LuΦ)dχ (3)

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Figure 2: Mode shapes for the first nine modes of the panel.

Θ =

∫χ

(LuΦ)T Sp(χ)eT (LφΦ)dχ (4)

whereχ is the domain of the structure,Φ are the eigenfunctions of the structure,Sp is thespatial aperture of the transducer,ρp is the density of the piezoelectric material,Lu is theelastic differential operator,κp is the flexural stiffness of the piezoelectric element,e is amatrix of piezoelectric material constants, andLφ is the electrical differential operator [1,10]. Each equation was solved numerically using the eigenfunctions obtained from thenormal modes analysis.

The dynamic equations for the piezostructure can be rewritten in state-variable formas,

x = Ax + Bu

y = Cx + Du(5)

where

x(t) =

[η(t)

˙η(t)

], u(t) =

[f

v

], (6)

A =

[0 I

−M−1K −M−1Cs

], B =

[0 0

M−1Bf M−1Θ

], (7)

C =[

0 Φ], D = [0], (8)

and

M = Ms + Mp, K = Ks + Kp. (9)

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Note that theC matrix is defined such that the output of the system is velocity. The soundpower radiated by the structure is calculated using a discrete representation of Rayleigh’sintegral, as described by Gibbs et al. [20]. The final structural-acoustic model contains138 states, which includes 60 structural modes and 6 radiation modes. While an excitationfilter describing the turbulent boundary layer pressure field could be combined with thestructural-acoustic model discussed above, this additional step would not add much valueto the analysis. Therefore, a spatially uncorrelated disturbance is used despite the fact thatit couples to a plate differently than a turbulent boundary layer excitation [21]. In thisstudy, the uncorrelated disturbance is modeled using 50 uniformly distributed randompoint loads with unit amplitude. These random point forces are included in the vectorf ,shown in Equation1.

3 RESULTS AND DISCUSSION

Three types of control approaches are investigated numerically using the stiffened paneldiscussed in the previous section: HAC, LAC, and a combined HAC/LAC approach.

Figure 3: Actuator/sensor configurations and control architectures.

3.1 HAC: Linear quadratic Gaussian control

The simulated results presented in this section are created using decentralized LQG con-trollers with the HAC configuration shown in Figure3(a). The transducers consist of acenter mounted 0.145 m×0.145 m piezoceramic actuator and five point velocity sensorson each bay. The outputs of the five velocity sensors, shown with circles in Figure3, are

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summed together and treated as a single sensor. Gibbs et al. [12] have shown that this typeof transducer configuration provides a good tradeoff between controller complexity andperformance for a single bay design.

LQG control is an optimal control strategy for systems with process and measurementnoise. The controller combines a minimum variance state estimate with an optimal regula-tor. In general, the LQG design is based on two parameters,β, the frequency-independentcontrol effort penalty, andα, the ratio of measurement to process noise. A control effortpenalty of10−6 is used for all of the simulations presented in this work. To determineα,we assume that the process noise enters coincident with the actuator and generates thesensor response. The amplitude of the frequency response function (FRF) between theactuator and sensors on one of the bays can then be used to determine a reasonable valuefor α. Figure4 shows the sensor response for one of the bays along with two differentmeasurement noise levels. Since the Kalman estimator is designed with the assumptionthat only the sensor response above the measurement noise is due to the plant model, thestate estimates in the regions where measurement noise dominates the sensor responsetend to have low gains [22]. Therefore, increasing the value ofα tends to decrease thegain of the state estimates and makes the controller more conservative. The measurementnoise levels shown in Figure4 areαi = 10−5 andαii = 10−6.

Figure 4: Comparison of process and measurement noise levels.

Figure 5 shows the radiated sound power from a panel excited by an uncorrelateddisturbance with unit amplitude. The independent controllers designed withαi = 10−5

achieve a 2.1 dB integrated reduction in the response between 100 and 800 Hz. Reducingα, which tends to make the controller more aggressive, has a negative impact on perfor-mance. The controllers designed withαi = 10−6 cause spillover and only achieve anintegrated reduction of 1.7 dB. As discussed earlier, the performance of LQG based con-trollers is limited by the accuracy of the control model. When the plant differs from thecontrol model, inaccurate state estimation can cause spillover of control energy and desta-bilize the closed-loop system [23]. In this simulation, the bays are structurally coupledand the controllers are designed independently. Therefore, each controller introduces newdynamics that are not included in the control model. As a result, the aggressive controlsystem causes spillover, as shown in Figure5.

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Figure 5: Decentralized LQG control using the actuator/sensor configurationshown in Figure3(a).

It should be noted that centralized control strategies as well as sequential loop closuretechniques can be used to account for the dynamics introduced by neighboring controllers.Unfortunately, centralized strategies are generally not scalable, and sequential loop clo-sure techniques may not be tolerant to actuator or sensor failures.

Despite the interaction between bays, the conservative controller,αi = 10−5, effec-tively attenuates the 300 Hz, 383 Hz, 421 Hz, and 443 Hz resonances without consider-able spillover. Unfortunately, the decentralized LQG control strategy does not adequatelyreduce the resonance associated with the global mode at 233 Hz. In addition, the HACsystem does not target the resonances associated with the third or fourth modes. Whilethese modes do not radiate as efficiently as the first or second structural modes, they stillcontribute to the total radiated sound power from the panel.

3.2 LAC: Direct output feedback

The LAC systems studied in this paper consist of piezoceramic actuators and point ve-locity sensors combined with direct output feedback. The transducer configurations areselected to target modes not emphasized by the HAC system. In particular, the LAC sys-tems are designed to target the 233 Hz, 336 Hz, and 360 Hz resonances.

Recently, Gardonio and Elliott [24] have shown that direct output feedback with pointsensors and triangular piezoceramic actuators positioned along the edges of a simply sup-ported plate offers good stability bounds. These stability bounds allow high feedbackgains, which can be used to add large amounts of damping to the plate. The first LACconfiguration studied in this work, shown in Figure3(b), uses a similar transducer config-uration consisting of two actuator/sensor pairs positioned on either side of the stiffener.The goal is to determine whether a triangular actuator and point sensor can be used alongthe flexible boundary to reduce the amplitude of the 233 Hz resonance.

The frequency response function between one of the actuator/sensor pairs is shown inFigure6. Since the system is minimum phase only up to 300 Hz, gains that attenuate the233 Hz resonance cause spillover in the 383 Hz resonance, as shown in Figure7. Theseresults indicate that the attractive collocation and duality features of the point sensor and

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triangular actuator can be lost when the boundary is flexible. This result is not surprisingsince a triangular actuator produces bending moments along its edges and transverse pointforces at the three vertices [25]. If the base of the triangle is positioned along arigidboundary, then the transverse point forces at its base will not couple to the plate, andcan therefore be neglected. The point sensor at the vertex opposite of the base edge willthen couple to the plate in much the same way as the actuator. If the stiffener isflexible,the transverse point forces at the base of the actuator can not be neglected. Therefore theactuator and sensor will not necessarily have good stability bounds, as shown in Figure6.

Figure 6: FRF for one ofthe transducer pairs shownin Figure3(b)

Figure 7: Direct output feed-back using the LAC configu-ration shown in Figure3(b).

The LAC transducer configuration, shown in Figure3(c), has good stability boundsdespite the flexible boundary. For this configuration, the summed output from four sen-sors is fed back to control a large rectangular patch that extends across both bays. Thefrequency response function is passive through 1200 Hz, and emphasizes the first struc-tural mode. The improved stability bounds are due to the size of the actuator and thelocation of the distributed sensors. As shown in Figure8, this LAC system reduces thefirst resonance by 12 dB and achieves a 1.5 dB integrated reduction from 100 to 800 Hz.

Figure 8: Direct output feedback using the LAC configuration shown in Figure3(c).

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Although the triangular actuator and point sensor are not effective along the stiffener,they can be used on the clamped edges of the panel. This improves the stability boundsof the LAC system since the bending moment and transverse point forces generated atthe clamped boundary can be neglected. The frequency response function for one of thetransducer pairs, shown in Figure9, is passive through 1500 Hz.

Due to the stability of this control arrangement, high feedback gains could be used toreduce the amplitude of the panel resonances as shown in Figure10. With only 4 triangularpiezoceramic patches and 4 point sensors, this LAC system attenuates the radiated soundpower by 3.4 dB integrated from 100 to 800 Hz. However, these transducer pairs do notefficiently couple to the first or second structural modes.

Figure 9: FRF for one ofthe transducer pairs shownin Figure3(d).

Figure 10: Direct outputfeedback using the LACconfiguration shown in Fig-ure3(d).

3.3 Combined HAC/LAC approach

Figure3(e) shows the combined HAC/LAC system, which consists of 8 sensors per bay,with piezoceramic actuators that cover approximately 20% of the surface area and add7% to the weight of the panel. For these simulations, the LQG controllers are designedusing plant responses that are identified after the low-authority feedback loops are closed.In general, the LAC system reduces the amplitude of the resonances. Adding damping tothe structure makes the effective plant easier to model and thus improves the computa-tional efficiency of the HAC system. Figure11 shows the open and closed-loop perfor-mance of LAC and HAC/LAC control systems. Combining the LAC systems shown inFigure3(c) and Figure3(d), reduces the radiated sound power by 4.6 dB integrated from100 to 800 Hz. The combined HAC/LAC system reduces the 300 Hz resonance by an ad-ditional 10 dB and achieves a 6.1 dB reduction in radiated sound power integrated from100 to 800 Hz.

Aside from the performance benefits discussed above, a combined HAC/LAC strat-egy can also be used to take advantage of specific characteristics of the disturbance. Forinstance, if a correlated reference signal is available, then the decentralized LQG portion

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Figure 11: Decentralized LQG and direct output feedback control using theHAC/LAC configuration shown in Figure3(e).

of the control architecture could be replaced with a hybrid control strategy that combinesboth feedback and feedforward techniques [1].

4 CONCLUSIONS

This paper presents simulated results for HAC, LAC, and combined HAC/LAC strategies.The HAC system presented in this paper is based on a decentralized LQG approach witha simple actuator/sensor configuration. Unfortunately, the performance of this design islimited by the fidelity of the control model. In addition, since only local information isavailable to design the controller, the global (multiple-bay) modes are not targeted by thecontrol system.

The LAC approach addresses the limitations of this particular HAC system. Specifi-cally, robust active damping is used to improve the overall system performance by cou-pling to modes not emphasized by the HAC design. Direct output feedback is used alongthe shared boundary to attenuate the resonance associated with the low frequency globalmode. This is implemented using a large rectangular piezoceramic actuator and the summedoutput of 4 velocity sensors. This actuator/sensor pair has good stability margins and ef-fectively attenuates the resonance associated with the first structural mode.

The transducer configuration presented by Gardonio and Elliott [24] consisting oftriangularly shaped piezoceramic patches and point velocity sensors appears to be partic-ularly well suited for a combined HAC/LAC control system. Grouping the actuator/sensorpairs along the boundaries of the panel achieves robust active damping without interfer-ing with the HAC transducers in the center of each bay. Simulated results indicate thatthis transducer configuration does not work well along a flexible boundary such as thestiffener, but is very effective if the boundaries are rigid.

The combined HAC/LAC system achieves good performance with relatively few ac-tuators and sensors. Simulated results indicate sound power reductions of 15 dB at thedominant 233 Hz resonance and more than 6 dB integrated from 100 to 800 Hz.

Although the HAC strategy presented in this work is based on a decentralized LQGapproach, in general, it could be selected based on the characteristics of the disturbance.

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If, for instance, a partially correlated and causal reference signal is available, then a hybridcontrol strategy could be implemented along with the LAC design. This will be the focusof future work.

5 ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of NASA Langley Research Center’sStructural Acoustics Branch and the National Institute of Aerospace.

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